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```LESSON
24.5
Properties and
Conditions for Kites
and Trapezoids
Name
HARDBOUND SE
PAGE 1005
BEGINS HERE
Class
24.5 Properties and Conditions for
Kites and Trapezoids
Essential Question: What are the properties of kites and trapezoids?
Resource
Locker
Exploring Properties of Kites
Explore
Common Core Math Standards
A kite is a quadrilateral with two distinct pairs of congruent consecutive sides.
_ _
_ _
_ _
In the figure, PQ ≅ PS, and QR ≅ SR, but QR ≇ QP.
The student is expected to:
G-SRT.5
Q
Use congruence and similarity criteria for triangles to solve problems and
to prove relationships in geometric figures.
T
P
Mathematical Practices
Measure the angles made by the sides and diagonals of a kite,
noticing any relationships.
Language Objective
The diagonals of a kite are perpendicular; the
diagonals of an isosceles trapezoid are congruent; a
kite has exactly one pair of congruent opposite
angles; an isosceles trapezoid has two pairs of
congruent base angles.
© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©Larry
Mulvehill/Corbis
Explain to a partner how to describe the properties of kites and
trapezoids.
Essential Question: What are the
properties of kites and trapezoids?
A
Use a protractor to measure ∠PTQ_
and ∠QTR in the figure. What do your results tell you
_
about the kite’s diagonals, PR and QS?
_
_
m∠PTQ = 90°, m∠QTR = 90°; the diagonals PR and QS are perpendicular.
B
Use a protractor to measure ∠PQR and ∠PSR in the figure. How are these opposite
angles related?
∠PQR ≅ ∠PSR
C
Measure ∠QPS and ∠QRS in the figure. What do you notice?
∠QPS ≇ ∠QRS, so only one pair of opposite angles in the kite are congruent.
D
Use a compass to construct your
own kite figure on a separate sheet
of paper. Begin by choosing a point
B. Then use your compass to choose
points A and C so that AB = BC.
E
Now change the compass
length and draw arcs from
both points A and C. Label
the intersection of the arcs
as point D.
C
B
D
Mark the intersection of the
diagonals as point E.
Module 24
E
D
A
A
be
ges must
EDIT--Chan
DO NOT Key=NL-A;CA-A
Correction
Lesson 5
1241
gh “File info”
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Date
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24.5 Pro and Trapezoids
Kites
Name
oids?
and trapez
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the prope
for triangl
What are
ity criteria
Question:
and similar
Essential
congruence figures.
G-SRT.5 Use
tric
of Kites
in geome
relationships
Properties
utive sides.
Exploring
ent consec
of congru
Explore
t pairs
_ _
two distinc
.
_
QP
with
≇
QR
_
lateral
_
a quadri
≅ SR, but
_ PS
A kite is
, and QR
PQ ≅
In the figure,
Q
T
HARDCOVER PAGES 10051018
Resource
Locker
and to prove
Turn to these pages to
find this lesson in the
hardcover student
edition.
R
P
S
kite,
nals of a
and diago
tell you
the sides
your results
s made by
the angle
. What do
Measure
onships.
in the figure
any relati
and ∠QTR
_
ndicular.
noticing
re ∠PTQ_
_
QS are perpe
ctor to measu _
QS?
Use a protra diagonals, PR and
nals PR and
the diago
kite’s
ite
TR = 90°;
about the
these oppos
90°, m∠Q
. How are
m∠PTQ =
the figure
∠PSR in
∠PQR and
re
ctor to measu
Use a protra
?
angles related
∠PSR
?
ruent.
∠PQR ≅
do you notice
are cong
What
kite
.
s in the
the figure
and
∠QRS in
site angle
the sides
∠QPS and
pair of oppo
Finally, draw
Measure
only one
ss
of the kite.
∠QRS, so
e the compa
diagonals
∠QPS ≇
Now chang
from
of the
draw arcs
intersection
uct your
length and
Label
Mark the
ss to constr
A and C.
as point E.
Use a compa on a separate sheet
both points
arcs
diagonals
figure
ction of the
a point
interse
ing
own kite
the
choos
e
Begin by
ss to choos
of paper.
as point D.
your compa
C
BC.
=
B. Then use
that AB
and C so
points A
D

y ⋅ Image
g Compan
Credits: ©Larry




Harcour
n Mifflin
© Houghto
Mulvehill/Corbis
t Publishin

B
C
C
D
E
A
B
B
A
A
Lesson 5
1241
Module 24
4L5.indd
62_U9M2
ESE3897
IN1_MNL
Lesson 24.5
Finally, draw the sides and
diagonals of the kite.
B
B
A
IN1_MNLESE389762_U9M24L5.indd 1241
1241
F
C
C
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo.
Ask students to describe the spider web and in
particular its geometrical properties. Then preview
the Lesson Performance Task.
R
S
MP.6 Precision
ENGAGE
Date
1241
4/18/14
8:49 PM
4/18/14 8:51 PM
G
HARDBOUND SE
Measure the angles of the kite ABCD you constructed in Steps D–F
and the measure of the angles formed by the diagonals. Are your
results the same as for the kite PQRS you used in Steps A–C?
PAGE 1006
BEGINS HERE
Yes, for kite ABCD, m∠AEB = 90° and m∠CEB = 90°, so, again, the diagonals of the kite
EXPLORE
Exploring Properties of Kites
are perpendicular. Also, for the constructed kite, ∠ABC ≇ ∠ADC, but ∠BAD ≅ ∠BCD, so,
again, one pair of opposite angles are congruent but the other pair are not.
INTEGRATE TECHNOLOGY
Students have the option of doing the kite activity
either in the book or online.
Reflect
1.
In the kite ABCD you constructed in Steps D–F, look at ∠CDE and ∠ADE. What do
you notice? _
Is this true for ∠CBE and ∠ABE as well? How can you state this in terms
of diagonal AC and the pair of non-congruent opposite angles ∠CBA and ∠CDA?
_
∠CDE ≅ ∠ADE; yes, ∠CBE ≅ ∠ABE; so, diagonal BD bisects the pair of non-
QUESTIONING STRATEGIES
congruent opposite angles ∠CBA and ∠CDA.
2.
What kind of triangles do the diagonals of a
kite form? Are any of these triangles
congruent? Explain. Right triangles; yes; there are
two pairs of congruent triangles by the HL Triangle
Congruence Theorem.
_
_
In the kite ABCD you constructed
_
_ in Steps D–F, look at EC and EA. What do you
notice? Is this true for EB and ED as well? Which diagonal is a perpendicular
bisector?
_ _
_ _
_
EC ≅ EA; no, EB ≇ ED; so, diagonal BD is the perpendicular bisector of
_
diagonal AC, but not vice versa.
Explain 1
EXPLAIN 1
Using Relationships in Kites
The results of the Explore can be stated as theorems.
Using Relationships in Kites
Four Kite Theorems
© Houghton Mifflin Harcourt Publishing Company
If a quadrilateral is a kite, then its diagonals are perpendicular.
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
If a quadrilateral is a kite, then one of the diagonals bisects the pair of
non-congruent angles.
If a quadrilateral is a kite, then exactly one diagonal bisects the other.
Q
P
T
R
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Communication
MP.3 Point out to students that in the previous
lesson they were introduced to the properties of
rectangles, rhombuses, and squares. Explain that in
this lesson, they will be given a quadrilateral and will
learn what conditions can be used to classify it as a
kite or a trapezoid. You may want to call on students
to read each theorem aloud. Then ask them to
explain the theorem in their own words. Challenge
students to come up with unique ways to explain the
theorems.
S
You can use the properties of kites to find unknown angle measures.
Module 24
1242
Lesson 5
PROFESSIONAL DEVELOPMENT
IN1_MNLESE389762_U9M24L5 1242
9/30/14 1:07 PM
Math Background
INTEGRATE TECHNOLOGY
A kite is a quadrilateral with two distinct pairs of congruent consecutive sides.
A trapezoid is a quadrilateral with at least one pair of parallel sides. These
definitions may not be the same as definitions in other textbooks. Such decisions
about definitions are somewhat arbitrary. Variations of definitions do not change
the facts of mathematics, but they do change the way the facts are expressed. The
decision to use an inclusive definition for trapezoid means that all parallelograms
share the properties of trapezoids. The decision to use an exclusive definition for
kite means that other quadrilaterals do not necessarily share the properties of kites.
Have students use geometry software to draw the
figures in some of the examples. This will allow them
to check their answers.
Properties and Conditions for Kites and Trapezoids
1242
QUESTIONING STRATEGIES
HARDBOUND SE
Example 1
PAGE 1007
BEGINS HERE
How do you use the properties of a kite to find
the measure of its angles? Since the diagonals
of a kite are perpendicular, they form right angles.
That makes the acute angles of the right triangles
complementary.

B
In kite ABCD, m∠BAE = 32° and
m∠BCE = 62°. Find each measure.
E
A

m∠CBE
Use angle relationships in △BCE.
C
D
m∠ABE
△ABE is also a right triangle.
Use the property that the diagonals of a kite are
perpendicular, so m∠BEC = 90°.
Therefore, its acute angles are complementary.
△BCE is a right triangle.
m∠ABE + m∠ BAE = 90 °
EXPLAIN 2
m∠BCE + m∠CBE = 90°
Substitute 32° for m∠ BAE , then solve for
m∠ABE.
Proving that Base Angles of Isosceles
Trapezoids Are Congruent
m∠CBE = 28°
Therefore, its acute angles are complementary.
Substitute 62° for m∠BCE, then solve for m∠CBE.
62° + m∠CBE = 90°
m∠ABE + 32 ° = 90 °
m∠ABE = 58 °
Reflect
3.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 You may want to review the Parallel Postulate
the sum of m∠ABE and m∠CBE.
Your Turn
4.
Determine m∠ADC in kite ABCD.
∠ADC ≅ ∠ABC , since exactly one pair of opposite angles are congruent.
m∠ADC = m∠ABC = m∠ABE + m∠CBE = 58° + 28° = 86°
Proving that Base Angles of Isosceles
Trapezoids Are Congruent
Explain 2
© Houghton Mifflin Harcourt Publishing Company
and the Corresponding Angles Theorem before you
present a plan for the proof. The Parallel Postulate
guarantees that there is a unique line through one
vertex of the trapezoid that is parallel to one leg of the
trapezoid. The segment determined by this line
creates a parallelogram, which in turn creates an
isosceles triangle (with congruent base angles). The
Corresponding Angles Theorem applies to the figure
because the lines are parallel, and the transitive
property will give congruent base angles.
From Part A and Part B, what strategy could you use to determine m∠ADC?
One pair of opposite angles in ABCD is congruent, so m∠ADC = m∠ABC. Also, m∠ABC is
A trapezoid is a quadrilateral with at least one pair of parallel sides. The pair of parallel sides of the trapezoid
(or either pair of parallel sides if the trapezoid is a parallelogram) are called the bases of the trapezoid.
The other two sides are called the legs of the trapezoid.
A trapezoid has two pairs of base angles: each pair consists of the two angles adjacent to one of the bases.
An isosceles trapezoid is one in which the legs are congruent but not parallel.
Isosceles trapezoid
Trapezoid
base
leg
leg
leg
base
base
base
leg
Module 24
1243
Lesson 5
COLLABORATIVE LEARNING
IN1_MNLESE389762_U9M24L5.indd 1243
Peer-to-Peer Activity
Ask students to work with a partner to make a physical model of a kite and of a
trapezoid with paper strips and brads. Have one student make a conjecture about
how to find the other three angle measures for the kite, and ask the other student
to make a conjecture about how to find the other three angle measures for the
trapezoid. Ask them to confirm each other’s conjecture by measuring the angles.
1243
Lesson 24.5
6/10/15 9:29 AM
HARDBOUND SE
Three Isosceles Trapezoid Theorems
PAGE 1008
BEGINS HERE
If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.
How is the isosceles triangle in this proof used
to show that the base angles of the isosceles
trapezoid are congruent? Sample answer: The
congruent sides of the isosceles triangle are used to
show that its base angles are congruent. Then the
proof establishes that the base angles of the
isosceles trapezoid are congruent by the transitive
property.
If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.
A trapezoid is isosceles if and only if its diagonals are congruent.
You can use auxiliary segments to prove these theorems.
Example 2
Complete the flow proof of the first Isosceles Trapezoid Theorem.
Given:
an isosceles
_ABCD
_ is_
_ trapezoid
with BC ǁ AD, AB ≅ DC.
B
C
Prove: ∠A ≅ ∠D
A
E
QUESTIONING STRATEGIES
D
BC || AD
Given
Draw CE || BA intersecting AD at E.
Parallel Postulate
∠A ≅ ∠CED
ABCE is a parallelogram.
Definition of parallelogram
Corresponding Angles Theorem
AB ≅ EC
Opposite sides of a
parallelogram are congruent.
AB ≅ DC
Given
EC ≅ DC
Substitution
∠CED ≅ ∠D
Isosceles Triangle Theorem
∠A ≅ ∠D
Reflect
5.
Explain how the auxiliary segment was useful in the proof.
Introducing the auxiliary segment breaks the trapezoid into familiar figures, a
parallelogram and an isosceles triangle. Then the properties of the simpler figures could
be used to prove the theorem about the more complex figure.
Module 24
1244
© Houghton Mifflin Harcourt Publishing Company
Transitive Property of Congruence
Lesson 5
DIFFERENTIATE INSTRUCTION
IN1_MNLESE389762_U9M24L5 1244
9/30/14 1:10 PM
Multiple Representations
Have students make a table of the properties of kites and trapezoids. Have them
list the properties of each in their own words and draw a diagram to represent
each. Then have students compare different types of quadrilaterals. For example,
ask: “How are kites and squares alike?” Sample answer: Both are quadrilaterals,
and both have perpendicular diagonals.
Properties and Conditions for Kites and Trapezoids
1244
6.
EXPLAIN 3
to ∠A and ∠C is supplementary to ∠D by the Same–Side Interior Angles Postulate. From
Example 2, ∠A ≅ ∠D. So ∠B ≅ ∠C because angles supplementary to ≅ angles are ≅
Using Theorems about Isosceles
Trapezoids
HARDBOUND SE
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Some math textbooks define a trapezoid as a
PAGE 1009
BEGINS HERE
Your Turn
7.
quadrilateral with exactly one pair of parallel sides.
Remind students that this definition is not used here.
Parallelograms are a subset of trapezoids because a
trapezoid has at least one pair of parallel sides, as
they are defined here. Students need to consider this
as they are using the theorems about isosceles
trapezoids to find segment lengths for trapezoids.
If you are trying to find the length of one part
of a diagonal of an isosceles trapezoid, what
information do you need? You need to know that
the diagonals of an isosceles trapezoid are
congruent.
Can the bases of a trapezoid be congruent?
Explain. Yes, if the trapezoid is also a
parallelogram.
Complete the proof of the second Isosceles Trapezoid Theorem: If a trapezoid has one pair
of base angles congruent, then the trapezoid is isosceles.
_ _
Given: ABCD is a trapezoid with BC ǁ AD, ∠A ≅ ∠D.
B
C
Prove: ABCD is an isosceles trapezoid.
A
E
_ _
D
_
_
_
BA so that CE
It is given that BC ǁ AD . By the Parallel Postulate , CE can be drawn parallel to
_
intersects AD at E. By the Corresponding Angles Theorem, ∠A ≅ ∠CED . It is given that ∠A ≅ ∠D ,
_
¯
∠CED ≅ ∠D . By the Converse of the Isosceles Triangle Theorem, CE
CD .
≅
so by substitution,
opposite sides
ABCE
By definition,
is a parallelogram. In a parallelogram,
are congruent, so
_
_
_ CD
_ CE
AB ≅
. By the Transitive Property. of Congruence, AB ≅
. Therefore, by definition,
ABCD is an isosceles trapezoid .
© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©Mariusz
Niedzwiedzki/Shutterstock
QUESTIONING STRATEGIES
The flow proof in Example 2 only shows that one pair of base angles is congruent.
Write a plan for proof for using parallel lines to show that the other pair of base
angles (∠B and ∠C) are also congruent.
Plan for Proof: In the isosceles trapezoid, the bases are parallel, so ∠B is supplementary
Explain 3
Using Theorems about Isosceles Trapezoids
You can use properties of isosceles trapezoids to find unknown values.
Example 3

Find each measure or value.
A railroad bridge has side sections that show isosceles trapezoids. The figure
ABCD represents one of these sections. AC = 13.2 m and BE = 8.4 m. Find DE.
B
C
E
A
D
Use the property that the diagonals are congruent.
_ _
AC ≅ BD
Use the definition of congruent segments.
AC = BD
Substitute 13.2 for AC.
13.2 = BD
Use the Segment Addition Postulate.
BE + DE = BD
Substitute 8.4 for BE and 13.2 for BD.
8.4 + DE = 13.2
Subtract 8.4 from both sides.
DE = 4.8
Module 24
1245
Lesson 5
LANGUAGE SUPPORT
IN1_MNLESE389762_U9M24L5 1245
Connect Vocabulary
Have students draw a kite and an isosceles trapezoid on poster board and label the
diagrams accordingly. For kites, include that their diagonals are perpendicular and
that they have one pair of congruent opposite angles. For isosceles trapezoids,
include that each pair of base angles is congruent and that the diagonals are
congruent.
1245
Lesson 24.5
9/30/14 2:01 PM
B
Find the value of x so that trapezoid EFGH is isosceles.
F
HARDBOUND SE
(2x2 + 21)
PAGE 1010
BEGINS HERE
G
H
E
(3x2 - 4)
For EFGH to be isosceles, each pair of base angles are congruent.
F
∠E ≅ ∠
F are congruent.
In particular, the pair at E and
m∠E = m∠ F
Use the definition of congruent angles.
2
Substitute 3x - 4 for m∠E and 2x + 21 for m∠ F .
.
3x 2 - 4 = 2x 2 + 21
2
2
Substract 2x from both sides and add 4 to both sides.
Take the square root of both sides.
x2 =
25
x=
5
or x =
-5
Your Turn
8.
In isosceles trapezoid PQRS, use the Same-Side
Interior Angles Postulate to find m∠R.
9.
Q
JL = 3y + 6 and KM = 22 − y. Determine the
value of y so that trapezoid JKLM is isosceles.
J
R
P
77°
K
M
L
For JKLM to be isosceles, its diagonals must
be congruent.
_ _
JL ≅ KM
S
Since PQRS is isosceles, each pair of base
angles must be congruent.
© Houghton Mifflin Harcourt Publishing Company
JL = KM
3y + 6 = 22 - y
∠P ≅ ∠Q; m∠P = m∠Q; 77° = m∠Q
4y = 16
Using the Same-Side Interior Angles
Postulate,
y=4
m∠Q + m∠R = 180°; 77° + m∠R = 180°
m∠R = 103°
Explain 4
Using the Trapezoid Midsegment Theorem
The midsegment of a trapezoid is the segment
whose endpoints are the midpoints of the legs.
Module 24
IN1_MNLESE389762_U9M24L5.indd 1246
HARDBOUND SE
PAGE 1011
BEGINS HERE
midsegment
1246
Lesson 5
4/18/14 8:50 PM
Properties and Conditions for Kites and Trapezoids
1246
Trapezoid Midsegment Theorem
EXPLAIN 4
The midsegment of a trapezoid is parallel to
each base, and its length is one half the sum
of the lengths of the bases.
_ _ _ _
XY ∥ BC , XY ∥ AD
Using the Trapezoid Midsegment
Theorem
B
X
D
You can use the Trapezoid Midsegment Theorem to find the length of the midsegment or a
base of a trapezoid.
In the formula for the length of the
midsegment, how do you know which
segment lengths to substitute where? Sample
answer: The midsegment length will be the length
inside the trapezoid connecting the midpoints of
the legs; the base lengths will be the lengths of the
parallel sides.
Find each length.
Example 4

E
In trapezoid EFGH, find XY.
X
F
12.5
10.3
H
Use the second part of the Trapezoid Midsegment Theorem.
Substitute 12.5 for EH and 10.3 for FG.
Students may have trouble keeping track of given
information, especially when algebraic expressions
are involved. Encourage students to color code
measures on each diagram. This may help students
identify the information needed for their calculations.

G
Y
1 (EH + FG)
XY = _
2
1 (12.5 + 10.3)
=_
2
= 11.4
Simplify.
AVOID COMMON ERRORS
L
In trapezoid JKLM, find JM.
Q
M
8.3
9.8
K
PAGE 1012
BEGINS HERE
© Houghton Mifflin Harcourt Publishing Company
P
HARDBOUND SE
Use the second part of the Trapezoid Midsegment Theorem.
Substitute
9.8 for PQ and
8.3 for
KL .
Multiply both sides by 2.
J
1
_
KL
PQ = (
+ JM)
2
1 8.3 + JM
9.8 = _
(
)
2
19.6 = 8.3 + JM
11.3 = JM
8.3 from both sides.
Subtract
Your Turn
10. In trapezoid PQRS, PQ = 2RS. Find XY.
PQ = 2RS
16.8 = 2RS
Q
Y
16.8
P
R
X
Module 24
IN1_MNLESE389762_U9M24L5.indd 1247
Lesson 24.5
Y
A
1 (BC + AD)
XY = _
2
QUESTIONING STRATEGIES
1247
C
S
8.4 = RS
1
XY = (PQ + RS)
2
1
= (16.8 + 8.4)
2
= 12.6
_
_
1247
Lesson 5
4/18/14 8:50 PM
Elaborate
ELABORATE
11. Use the information in the graphic organizer to complete the Venn diagram.
Rectangle
Four right angles
Parallelogram
Two pairs of
parallel sides
Rhombus
Four congruent sides
Quadrilateral
Square
Four right angles and
four congruent sides
QUESTIONING STRATEGIES
Why can’t theorems about the base angles of
an isosceles trapezoid be written as
biconditionals while theorems about the diagonals
can? Sample answer: One starts with an isosceles
trapezoid and results in finding two pairs of base
angles congruent; the other starts with a trapezoid
and one pair of congruent base angles and reasons
the trapezoid is isosceles. So, the hypothesis and
conclusion of the two theorems are not the reverse
of each other.
Trapezoid
At least one pair of
parallel sides
Kite
Two distinct pairs of
congruent sides
Quadrilateral
Parallelogram
Two pairs of
parallel sides
Rectangle
Four right angles
Square
Rhombus
Four right angles and
four congruent sides Four congruent sides
SUMMARIZE THE LESSON
Kite
Two distinct pairs of
congruent sides
Have students list the properties of kites and
trapezoids and the difference between a
trapezoid and an isosceles trapezoid. Sample
answer: The diagonals of a kite are perpendicular;
the diagonals of an isosceles trapezoid are
congruent; a kite has exactly one pair of congruent
opposite angles; an isosceles trapezoid has two
pairs of congruent base angles. An isosceles
trapezoid has legs that are congruent but not
parallel.
Trapezoid
At least one pair of
parallel sides
HARDBOUND SE
12. Discussion The Isosceles Trapezoid Theorem about congruent diagonals is in the form
PAGE 1013
of a biconditional statement. Is it possible to state the two isosceles trapezoid theorems
BEGINS HERE
about base angles as a biconditional statement? Explain.
No; the hypotheses and conclusions are not reverses. One has “isosceles trapezoid” as a
hypothesis and “each pair of base angles are ≅” as a conclusion; the other has “a trapezoid
and one pair of ≅ base angles” as a hypothesis and “isosceles trapezoid” as a conclusion.
© Houghton Mifflin Harcourt Publishing Company
What can you conclude about all parallelograms? Possible answer: All
parallelograms are quadrilaterals and all parallelograms are also trapezoids.
13. Essential Question Check-In Do kites and trapezoids have properties that are related
to their diagonals? Explain.
Yes; the diagonals of a kite are perpendicular, while the diagonals of an
isosceles trapezoid are congruent.
Module 24
IN1_MNLESE389762_U9M24L5.indd 1248
1248
Lesson 5
4/18/14 8:50 PM
Properties and Conditions for Kites and Trapezoids
1248
Evaluate: Homework and Practice
EVALUATE
B
In kite ABCD, m∠BAE = 28° and
m∠BCE = 57°. Find each measure.
E
A
C
D
ASSIGNMENT GUIDE
1.
Concepts and Skills
Practice
Explore
Exploring Properties of Kites
Exercises 16–18
Example 1
Using Relationships in Kites
Exercises 1–4
Example 2
Proving that Base Angles of
Isosceles Trapezoids Are Congruent
Exercises 5–6
Example 3
Using Theorems about Isosceles
Trapezoids
Exercises 7–10
Example 4
Using the Trapezoid Midsegment
Theorem
Exercises 11–14
4.
m∠ADC
m∠ABC = m∠ABE + m∠CBE
∠ADC ≅ ∠ABC
= 95°
m∠ADC = 95°
m∠ADC = m∠ABC
= 62° + 33°
Using the first and second Isosceles Trapezoid Theorems, complete the proofs of each part of the third
Isosceles Trapezoid Theorem: A trapezoid is isosceles if and only if its diagonals are congruent.
Prove part 1: If a trapezoid is isosceles, then its diagonals are congruent.
B
Given:
ABCD
_ _
_ is an
_isosceles trapezoid with
BC ∥ AD , AB ≅ DC .
_ _
Prove: AC ≅ DB
A
C
E
F
D
_ _
,
It is given that AB ≅ DC . By the first Trapezoid Theorem, ∠BAD ≅ ∠CDA
― ―
and by the Reflexive Property of Congruence, AD ≅ AD . By the SAS Triangle
_ _
Congruence Theorem, △ABD ≅ △DCA, and by CPCTC , AC ≅ DB .
Exercise
IN1_MNLESE389762_U9M24L5 1249
Lesson 24.5
m∠CBE = 33°
m∠ABC
Module 24
1249
57° + m∠CBE = 90°
m∠ABE = 62°
3.
m∠CBE
m∠BCE + m∠CBE = 90°
m∠ABE + m∠BAE = 90°
© Houghton Mifflin Harcourt Publishing Company
realize how to begin solving the problem. Point out
that they must first determine if the figure is a kite, a
trapezoid, or an isosceles trapezoid. Then suggest
that they redraw each figure, marking known
properties for that type of quadrilateral.
2.
m∠ABE + 28° = 90°
5.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 For some exercises, some students may not
m∠ABE
Lesson 5
1249
Depth of Knowledge (D.O.K.)
Mathematical Practices
9/30/14 2:06 PM
1–4
1 Recall of information
MP.4 Modeling
5–6
2 Skills/Concepts
MP.3 Logic
7–19
2 Skills/Concepts
MP.4 Modeling
20–21
2 Skills/Concepts
MP.4 Modeling
22–24
2 Skills/Concepts
MP.2 Reasoning
25
3 Strategic Thinking
MP.3 Logic
26
3 Strategic Thinking
MP.3 Logic
6.
HARDBOUND SE
Prove part 2: If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.
_ _
_ _
Given: ABCD is a trapezoid with BC ∥ AD and diagonals AC ≅ DB .
Prove: ABCD is an isosceles trapezoid.
B
A
E
F
Statements
_ _
2. BE ∥ CF
_ _
3. BC ∥ AD
4. BCFE is a parallelogram.
_ _
5. BE ≅ CF
_ _
6. AC ≅ DB
Reasons
1. There is only one line through a given point
perpendicular to a given line, so each auxiliary
line can be drawn.
2. Two lines perpendicular to the same line are
parallel.
3. Given
4. Definition of parallelogram (Steps 2, 3)
5. If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
6. Given
7. Definition of perpendicular lines
8. △BED ≅ △CFA
8. HL Triangle Congruence Theorem (Steps 5–7)
9. ∠BDE ≅ ∠CAF
9. CPCTC
10. ∠CBD ≅ ∠BDE, ∠BCA ≅ ∠CAF
10. Alternate Interior Angles Theorem
11. ∠CBD ≅ ∠BCA
11. Transitive Property of Congruence (Steps 9, 10)
12.
12. Given
14. SAS Triangle Congruence Theorem
(Steps 12, 13)
15. ∠BAC ≅ ∠CDB
15. CPCTC
17. ABCD is isosceles.
Module 24
IN1_MNLESE389762_U9M24L5.indd 1250
© Houghton Mifflin Harcourt Publishing Company
13. Reflexive Property of Congruence
14. △ABC ≅ △DCB
16. ∠BAD ≅ ∠CDA
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Encourage students to develop their own
work patterns when they analyze the many types of
quadrilaterals in the exercises, especially when
algebraic expressions are involved. For example, they
may want to color code the measures on each
diagram to help them identify the information
needed for their calculations.
D
7. ∠BED and ∠CFA are right angles.
_ _
13. BC ≅ BC
BEGINS HERE
C
_ _
_ _
1. Draw BE ⊥ AD and CF ⊥ AD.
― ―
AC ≅ DB
PAGE 1014
16. Angle Addition Postulate
17. If a trapezoid has one pair of base angles
congruent, then the trapezoid is isosceles.
1250
Lesson 5
4/18/14 8:50 PM
Properties and Conditions for Kites and Trapezoids
1250
HARDBOUND SE
Use the isosceles trapezoid to find each measure or value.
PAGE 1015
BEGINS HERE
7.
LJ = 19.3 and KN = 8.1. Determine MN.
K
Find the positive value of x so that trapezoid
PQRS is isosceles.
8.
L
Q
(2x2 + 5)
J
2x 2 + 5 = x 2 + 30
x 2 = 25
S
M
―
L J ≅ KM ; L J = KM; 19.3 = KM;
KN + MN = KM; 8.1 + MN = 19.3; MN = 11.2
9.
m∠Q = m∠P
R
N
―
∠Q ≅ ∠P
In isosceles trapezoid EFGH, use the Same-Side
Interior Angles Postulate to determine m∠E.
x=5
(x2 + 30)
P
10. AC = 3y + 12 and BD = 27 - 2y. Determine
the value of y so that trapezoid ABCD is
isosceles.
_ _
AC ≅ BD
A
E
F
H
G
AC = BD
3y + 12 = 27 - 2y
5y = 15
B
137°
y=3
∠G ≅ ∠H; m∠G = m∠H; 137° = m∠H
C
m∠E + m∠H = 180°; m∠E + 137° = 180°;
m∠E = 43°
D
Find the unknown segment lengths in each trapezoid.
12. In trapezoid EFGH, find FG.
© Houghton Mifflin Harcourt Publishing Company
11. In trapezoid ABCD, find XY.
13.3
D
C
A
XY =
X
Y
17.9
E
B
2
2
13. In trapezoid PQRS, PQ = 4RS. Determine XY.
G
Q
Y
7.4 = 2JK
3.7 = JK
Q
18.4
Module 24
IN1_MNLESE389762_U9M24L5.indd 1251
7.4
J
S
_
_1 (JK + LM)
2
1
7.4 = _(3.7 + LM)
PQ =
K
PQ = 4RS; 18.4 = 4RS; 4.6 = RS
1
1
XY = (PQ + RS) = (18.4 + 4.6) = 11.5
2
2
_
17.0 = FG
PQ = 2JK
M
X
2
39.4 = 22.4 + FG
14. In trapezoid JKLM, PQ = 2JK. Determine LM.
Q
P
Lesson 24.5
19.7
H
_1 (DC + AB) = _1 (13.3 + 17.9) = 15.6
R
1251
F
22.4
_1 (EH + FG)
2
1
19.7 = _(22.4 + FG)
PQ =
P
P
L
2
14.8 = 3.7 + LM
11.1 = LM
1251
Lesson 5
3/3/16 10:31 PM
15. Determine whether each of the following describes a kite or a trapezoid. Select the
correct answer for each lettered part.
A. Has two distinct pairs of congruent
consecutive sides
kite
trapezoid
B. Has diagonals that are perpendicular
kite
trapezoid
C. Has at least one pair of parallel sides
kite
kite
trapezoid
trapezoid
kite
trapezoid
D. Has exactly one pair of opposite angles
that are congruent
E. Has two pairs of base angles
AVOID COMMON ERRORS
HARDBOUND SE
16. Multi-Step Complete the proof of each of the four Kite Theorems. The proof of
each of the four theorems relies on the same initial reasoning, so they are presented
here in a single two-column proof.
B
_
_
_
_
Given: ABCD is a kite, with AB ≅ AD and CB ≅ CD .
_
_
Prove: (i) AC ⊥ BD ;
E
A
C
ABC ≅ ∠ADC;
(ii) ∠_
BAD and ∠BCD;
(iii) AC
_ bisects ∠
_
(iv) AC bisects BD.
PAGE 1016
BEGINS HERE
D
Statements
_ _ _ _
1. AB ≅ AD, CB ≅ CD
_
_
2. AC ≅ AC
Reasons
1. Given
2. Reflexive Property of Congruence
3. SSS Triangle Congruence Theorem
(Steps 1, 2)
4. ∠BAE ≅ ∠DAE
4. CPCTC
_ _
5. AE ≅ AE
5. Reflexive Property of Congruence
6. △ABE ≅ △ADE
6. SAS Triangle Congruence Theorem
(Steps 1, 4, 5)
7. ∠AEB ≅ ∠AED
7. CPCTC
8. If two lines intersect to form a linear pair
of congruent angles, then the lines are
perpendicular.
9. ∠ABC ≅ ∠ADC
9. CPCTC
(Step 3)
10. ∠BAC ≅ ∠DAC and ∠BCA ≅ ∠DCA
10. CPCTC
(Step 3)
_
11. AC bisects ∠BAD and ∠BCD.
11. Definition of angle bisector
12.
_
BE
≅
_
DE
_
_
13. AC bisects BD.
Module 24
IN1_MNLESE389762_U9M24L5 1252
© Houghton Mifflin Harcourt Publishing Company
3. △ABC ≅ △ADC
_ _
8. AC ⊥ BD
Some students may have trouble understanding how
a trapezoid is isosceles if and only if its diagonals are
congruent. Ask them to break up this biconditional
statement into two statements to prove. They should
see that if the diagonals are congruent, they can use
triangle criteria to show congruent triangles and then
use corresponding parts of congruent figures to show
the trapezoid is isosceles. Conversely, they can start
with an isosceles trapezoid to show triangles are
congruent and then use corresponding parts of
congruent figures to show the diagonals are
congruent.
12. CPCTC (Step 6)
13. Definition of segment bisector
1252
Lesson 5
9/30/14 2:09 PM
Properties and Conditions for Kites and Trapezoids
1252
17. Given: JKLN is a parallelogram. JKMN is an isosceles
trapezoid.
K
J
Prove: △KLM is an isosceles triangle.
N
_ _
M
1. JKLN is a parallelogram. (Given); 2. KL ≅ NJ
(Opposite sides of a parallelogram are congruent.);
_ _
3. JKMN is an isosceles trapezoid. (Given); 4. NJ ≅ MK (Definition of
isosceles
trapezoid);
_ _
5. KL ≅ KM (Transitive Property of Congruence); 6. △KLM is an isosceles
triangle. (Definition of isosceles triangle)
L
Algebra Find the length of the midsegment of each trapezoid.
18.
19.
12
4x
3y - 7
y+6
6x
_
1
4x = (12 + 6x)
2
8x = 12 + 6x
x = 6, so 4x = 4(6) = 24
HARDBOUND SE
PAGE 1017
BEGINS HERE
y+3
_(
)
1
y+6=
(y + 3) + (3y - 7)
2
2(y + 6) = (y + 3) + (3y - 7)
2y + 12 = 4y - 4
8 = y, so y + 6 = 8 + 6 = 14
20. Represent Real-World Problems A set of shelves fits an attic room with one
sloping wall. The left edges of the shelves line up vertically, and the right edges line up
along the sloping wall. The shortest shelf is 32 in. long, and the longest is 40 in. long.
Given that the three shelves are equally spaced vertically, what total length of shelving
is needed?
The shelves form a trapezoid and the middle shelf forms its midsegment.
1
middle shelf: (32 in. + 40 in.) = 36 in. Total: 32 + 36 + 40 = 108 in.
2
© Houghton Mifflin Harcourt Publishing Company
_
C
21. Represent Real-World Problems A common early stage in making an
origami model is known as the kite. The figure shows a paper model at this
stage unfolded.
The folds create four geometric kites. Also, the 16 right triangles adjacent to
the corners of the paper are all congruent, as are the 8 right triangles adjacent
to the center of the paper. Find the measures of all four angles of the kite
labeled ABCD (the point A is the center point of the diagram). Use the facts
that ∠B ≅ ∠D and that the interior angle sum of a quadrilateral is 360°.
B
A
D
The 8 congruent central triangles are isosceles right triangles; so, m∠A =
2(45)° = 90°. The four angles at each corner of the paper are congruent,
1
so m∠BCA = m∠DCA = (90°) = 22.5° and m∠C = 22.5° + 22.5° = 45°.
4
Since ∠B ≅ ∠D, m∠B = m∠D = x°; then:
90 + x + 45 + x = 360°, 135 + 2x = 360, and x = 112.5.
So m∠B = m∠D = 112.5°.
_
Module 24
IN1_MNLESE389762_U9M24L5.indd 1253
1253
Lesson 24.5
1253
Lesson 5
4/18/14 8:49 PM
22. Analyze Relationships The window frame is a regular octagon. It is made
from eight pieces of wood shaped like congruent isosceles trapezoids. What are
m∠A, m∠B, m∠C, and m∠D in trapezoid ABCD?
_
_
Extend BA and CD to their intersection E. Since
1
m∠E = (360°) = 45°, and since ∠B ≅ ∠C,
8
m∠B + m∠C + m∠E = 180°
_
B
C
A
D
2m∠B + 45° = 180°
2m∠B = 135°
m∠B = 67.5°
m∠C = 67.5°
By the Same-Side Int. ∠s Post., m∠A + m∠B = 180°,
so m∠A = 180° - 67.5° = 112.5°.
Since ∠A ≅ ∠D, m∠D = 112.5°.
and m∠ADE
23. Explain the Error In kite ABCD, m∠BAE = 66° _
_= 59°.
Terrence is trying to find m∠ABC. He knows that BD bisects AC, and
that therefore △AED ≅ △CED. He reasons that ∠ADE ≅ ∠CDE, so that
m∠ADC = 2(59°) = 118°, and that ∠ABC ≅ ∠ADC because they are opposite
angles in the kite, so that m∠ABC = 118°. Explain Terrence’s error and
describe how to find m∠ABC .
Terrence mistakenly reasoned that ∠ABC ≅ ∠ADC; only one pair
of opposite angles in a kite are congruent, and they are adjacent
to the bisected diagonal, not the bisecting diagonal. To find
A
m∠ABC: Since ∠BAE and ∠ABE are complementary, m∠ABE can
be found by 90° - 66° = 24°. Then, since △AEB ≅ △CEB so that
∠ABE ≅ ∠CBE, m∠ABC is twice m∠ABE, or 2(24°) = 48°.
B
E
C
D
24. Complete the table to classify all quadrilateral types by the rotational symmetries and
line symmetries they must have. Identify any patterns that you see and explain what
these patterns indicate.
Angle of Rotational
Symmetry
© Houghton Mifflin Harcourt Publishing Company
Quadrilateral
Number of Line
Symmetries
kite
none
non-isosceles trapezoid
none
0
isosceles trapezoid
none
1
parallelogram
180°
0
rectangle
180°
2
rhombus
180°
2
square
90°
4
1
The quadrilaterals with rotational symmetry are parallelograms and special cases of
parallelograms; the more restricted cases of quadrilaterals tend to have more line
symmetries, up to the square with 4; there are two pairs of quadrilateral types with the
same symmetries, kites and isosceles trapezoids, and rectangles and rhombuses.
Module 24
IN1_MNLESE389762_U9M24L5 1254
1254
Lesson 5
9/30/14 2:11 PM
Properties and Conditions for Kites and Trapezoids
1254
JOURNAL
Have students draw an isosceles trapezoid. Have
them label the angle measures in terms of x and write
a justification for each measure.
HARDBOUND SE
H.O.T. Focus on Higher Order Thinking
PAGE 1018
BEGINS HERE
25. Communicate Mathematical Ideas Describe the properties that rhombuses and
kites have in common, and the properties that are different.
Rhombuses and kites both have pairs of congruent consecutive sides, but in a rhombus, this
is because all four sides are congruent. In a kite, two distinct pairs of consecutive sides are
congruent. The diagonals of both types of quadrilaterals are perpendicular. In a kite, exactly
one diagonal is bisected by the other, while in a rhombus, each diagonal bisects the other.
Finally, in a kite, exactly one pair of opposite angles are congruent, while both pairs of
opposite angles of a rhombus are congruent.
B
26. Analyze Relationships In kite ABCD, triangles
B
B'
ABD and CBD
can
be
rotated
and
translated,
_
_
identifying AD with CD and joining the remaining
pair of vertices, as shown in the figure. Why is this
process guaranteed to produce an isosceles trapezoid?
Suggest a process guaranteed to produce a kite from
an isosceles trapezoid, using figures to illustrate your
A
process.
C
© Houghton Mifflin Harcourt Publishing Company
A = D'
Module 24
IN1_MNLESE389762_U9M24L5.indd 1255
1255
Lesson 24.5
D = C'
Sample answer for the case of producing an
D
isosceles trapezoid from a kite where the
B
congruent angles are obtuse: By the
kite_and _
the
_ _ of a_
_definition
Reflexive
of Congruence, BA ≅ BC, AD ≅ CD, BA ≇ AD,
_ Property
_
and BD ≅ BD
_ and translate the triangles BAD and BCD so
_. Rotate
coincide, to produce the quadrilateral BB’DA
that sides AD and
_ CD _
_
and B'F_
perpendicular to the line containing AD at
shown. Draw BE_
points E and F. BE and B'F are parallel because they are perpendicular
to the same line. △BAD ≅ △B'C'D' by SSS, so ∠BAD ≅ ∠B'C'D'
because corresponding parts of congruent figures are congruent.
Then ∠BAE ≅ ∠B'C'F because supplements of congruent angles
E
A = D’
D = C’
are congruent.
_ _∠BEA ≅ ∠B'FC' because they are both right angles.
_ _
Also, BA ≅ B'C' (from the kite), so △BAE ≅ △B'C'F by AAS. So, BE ≅ B'F because corresponding
parts of congruent figures are congruent. Quadrilateral BB'FE is a parallelogram, because
_one
pair of
_and congruent. By the definition of a parallelogram, BB'
_parallel
_ opposite sides are
and EF (which includes AD (or C'D')) are parallel, so BB'DA is a trapezoid because it has a pair of
‹ ›
−
parallel sides. Consider the lines BA and B'C' cut by transversal EF . Then ∠BAE and ∠B'C'D' are
corresponding angles. Because ∠BAD is obtuse, its supplement, ∠BAE, is_
acute. But ∠B'C'D' is_
obtuse, which means that ∠BAE and ∠B'C'D' cannot be congruent. Thus, BA is not parallel to B'C'
and BB'DA is not a parallelogram. A trapezoid that is not a parallelogram but has congruent legs
is isosceles, so BB'DA is an isosceles trapezoid.
L = K'
Sample
an isosceles
trapezoid,
K
L
_ answer:
_ JKLM
_ is_
_
with JK ≅ ML but JK ≇ KL or JM. Rotate and _
translate
_ triangles JKL and MLK so that sides JL
and M'K' coincide. This can be done because the
M
diagonals of an isosceles trapezoid are congruent. J
K
The resulting figure
has
two
distinct
pairs
of
_ _
_ _
congruent sides, JK ≅ M'L' (given) and KL ≅ L'K'
(Reflexive Property of Congruence), so the figure
is a kite.
1255
B’
F
L'
J = M'
Lesson 5
6/10/15 9:36 AM
Lesson Performance Task
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 In the model of the spider web, CD = 6 cm
This model of a spider web is made using only isosceles triangles and isosceles trapezoids.
and BE = 3 cm. Without referring to the
circumference of either the outer or the inner circle,
explain how you know that the radius of the inner
circle is half the radius of the outer circle. By the
converse of the Triangle Midsegment Theorem, B is
the midpoint of ¯
AC. So, ¯
AB, a radius of the inner
circle, has half the measure of ¯
AC, a radius of the
outer circle.
A
B
E
C
D
a. All of the figures surrounding the center of the web are
congruent to figure ABCDE. Find m∠A. Explain how you
found your answer.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
_
MP.1
Explain how you know that BE is parallel
_
b. Find m∠ABE and m∠AEB.
c. Find m∠CBE and m∠DEB.
d. Find m∠C and m∠D.
a. 22.5°; 16 triangles congruent to △ABE surround the center of the web, making up
a total of 360°. 360° ÷ 16 = 22.5°
to CD. Sample answer: m∠ABE = 78.75° = m∠C, so
∠ABE ≅ ∠C. So, ¯
BE ǁ ¯
CD because corresponding
angles ∠ABE and ∠C are congruent.
© Houghton Mifflin Harcourt Publishing Company
b. m∠A = 22.5° and △ABE is an isosceles triangle. So,
∠ABE ≅ ∠AEB and m∠ABE = m∠AEB
180° = m∠ABE + m∠AEB + 22.5°
180° = m∠ABE + m∠ABE + 22.5°
157.5° = 2 ⋅ m∠ABE
78.75° = m∠ABE
Thus, each angle measures 78.75°.
c. Because the angles form a linear pair, then m∠ABE + m∠CBE = 180°.
So 78.75° + m∠CBE = 180° and m∠CBE = 101.25°.
In an isosceles trapezoid, base angles are congruent, so m∠DEB = 101.25°.
― ―
d. In isosceles trapezoid CBED, BE ‖ CD. Then corresponding angles are congruent and
∠ABE ≅ ∠C and ∠AEB ≅ ∠D. So m∠C = m∠D = 78.75°.
Module 24
1256
Lesson 5
EXTENSION ACTIVITY
IN1_MNLESE389762_U9M24L5 1256
The Lesson Performance Task deals with triangles and trapezoids in spider webs.
Have students research other examples that illustrate quadrilaterals in the animal
world, the plant world, or both. For each example students find, they should make
a sketch, identify the quadrilateral, and provide additional information they feel is
relevant, particularly as it relates to geometry.
10/17/14 11:31 PM
Scoring Rubric
2 Points: Student correctly solves the problem and explains his/her reasoning.
1 Point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 Points: Student does not demonstrate understanding of the problem.
Properties and Conditions for Kites and Trapezoids
1256
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